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The block caving mining method is used to mine large orebodies in weak rock types. The layout for block caving requires that a considerable amount of development is carried out on the extraction drives and drawpoints at the bottom of the caving block. The development further weakens the rock mass, and stability problems arise in the extraction drives and extraction pillars. This paper describes the application of three-dimensional boundary element models in analysing alternative extraction level layouts for a block caving mine. The objectives were to establish whether the use of lateral pillars between drawpoints was warranted, and whether the lateral pillars should be offset or not. The stability of the rock mass within the pillars as well as the stability of the extraction drives were taken into account. The results showed that after undercutting, the horizontal and vertical stresses in the extraction level would be reduced. The lateral pillars would significantly improve the stability of the extraction pillars and extraction drives. There would be a minor improvement in stability if the lateral pillars were offset.

Jan 1, 1987

By J. P. Hugo

Nuclear energy - its interactions with mining and metallurgy - PRESIDENTIAL ADDRESS given by J. P. Hugo Pr. Eng., B.Sc (Eng.) (Rand), Ph.D. (Sheffield) Energy is the life blood of the modern technological era in which we live and it was, I think, Benjamin Franklin who said that the manipulation of energy was the basis of all science and technology. This is perhaps an oversimplification, but there can be little doubt that our life today, and the improved standard of living we aspire to, are governed almost entirely by our ability to generate and manipulate energy. This ability has made it possible for Man - each specimen a feeble twentieth of a horsepower - to travel to the moon, to recover valuable minerals from the depths of the earth, to do arithmetic at unbelievable speed, to lessen the drudgery of manual labour. For these and virtually every aspect of our life we are dependent on energy from external sources. It is, therefore, not surprising that the per capita consumption of energy, and more particularly electricity, has become one of the most reliable criteria by which to measure the standard of living and technological advancement of a country.

By M. Akyüzlü

The equilibrium between carbon-saturated Fe-Cr-Si-C alloys and SiO2-CaO-MgO-A2O3-CrO2-FeOy slags was studied under atmospheres of argon and carbon monoxide at temperatures of 1500 and 1600°C. The equilibrium distribution of chromium, silicon, and iron between the slag and the metal phases was determined for different compositions of both phases under different atmospheres and temperatures. It is shown that silicon transfer from the slag to the metal increases with increasing temperature, decreasing carbon monoxide partial pressure, decreasing carbon content of the metal, increasing chromium-to-iron ratio of the metal, increasing silica content of the slag, and decreasing lime-to-alumina ratio of the slag. Some activity calculations were made for the experiments conducted at 1600OC in a carbon monoxide atmosphere. It is also shown that a decrease in the iron content and an increase in the slag basicity decrease the chromium content of the slag. A major proportion of the chromium present in the slag was found to be in the divalent state under the highly reducing conditions prevailing in the experiments. The chromous capacities of these slags were calculated and were found to decrease with increasing basicity.

Jan 1, 1992

By O. Fenn

Earlier work had shown that the use of water jets at moderate pressure enhances mechanical cutting in hard rock. To assess the potential of this technique for use in conjunction with free-rolling cutters, a series of laboratory tests was conducted with disc and button cutters to determine the effect of variations in the jet and cutting parameters on the cuter performance. In these tests it was found that the use of water jets at pressure in the range 5 to 40 MPa can reduce the forces on a free-rolling cuter by 40 per cent, which represents a significant improvement in cutting performance. The reduction in thrust force was found to be dependent on jet pressure: the greater the pressure, the greater the reduction. The reduction in rolling force was independent of jet pressure over the range investigated. The results indicate that no additional improvements in cutting performance of any significance are to be gained by an increase in jet pressure above 40 MPa. The most effective jet configuration for the reduction of cuter forces was found to be four coherent water jets, two lateraly disposed on each side of the cuter and directed at the arc of contact between the cutting edge and the rock.

Jan 1, 1987

By S. K. Mahata, P. G. Suresh Kumar, B. B. Singh

Improvement of post tap hole practices is achievable by mechanized handling of melt during the production of Ferro Chrome, resulting in higher recovery of chromium. The paper deals with modern practices adopted in the production of ferro-chrome since past two decades. The methods and practices deal with various areas : a) Improved method of operating the tap hole by selection of right quality of refractory to maintain the life of tap hole. b) Collection of melt in different types of receptacles like refractory brick lined mild steel ladles, cast steel ladles, cast iron pans being chosen for cost effective and efficient handling. c) Separation of slag and metal, casting of metal, crushing, sizing and stock piling etc. d) Slag disposal and recovery of entrapped metal. The drive behind the above development is continuous improvement, which results in higher chromium recovery and reduces loss of metal in slag by completely mechanized method so as to be at the same level with producers of ferro-chrome in other advanced countries , in the emerging global economy.

Jan 1, 2007

By J. C. M. De Souza

The Samarco pipeline, the longest in the world for iron ore slurry transportation, started operation in 1977 to transport concentrate from the Germano facilities, in the state of Minas Gerais, to the pelletizing plant, located 396 km away, in the state of Espirito Santo, on the south-east coast of Brazil. Designed for a nominal capacity of 12 million tons per year, the pipeline reached 15.5 million/year due to operational improvement and, in late 2005, it was resized to transport 16.5 million tons per year, with the installation of a booster station. In order to ensure the accurate performance of the system, a pipeline cleaning programme was carried out and, since it has been 29 years from the operation start-up, precise knowledge of the pipe condition would be fundamental to guarantee the success of the project. The cleaning programme demonstrated the potential of operational gains from this initiative, and its effect on the corrosion control and future measures, which should be permanently adopted to maintain the improved capacity. The intelligent pig runs provided essential data used to establish a pipeline integrity programme that allowed Samarco to develop a pipe repair plan based on the severity and significance of the features and to implement a future pipe replacement plan.

Jan 1, 2007

Actual observations of rockburst damage, together with some theoretical considerations, strongly suggest that conventional approaches to the design of tunnel support are inappropriate where there is a significant seismic risk. The present widely used support systems, which are based on stiff, fully grouted re-bars, are inadequate where rock displacement velocities exceed 1 m/s. It is shown by means of a simple conceptual model that yielding of the support elements is essential to ensure their survival when they are subject to the high strain rates typical of severe rockbursts. The model proposes that the total load in the element is an impulsive tensile force imposed and sustained by a single 'tributary' block ejected from the tunnel wall. The stability of the system is determined from a comparison of the kinetic energy developed by the block with the energy of deformation absorbed by a grouted steel tendon. A novel test procedure was developed for the validation of this approach, and for an examination of the energy-absorbing capacity of stiff and yielding tendons. The results of an initial series of tests showed that the recently developed cone-bolt is an effective yielding device. It absorbs twenty times more impulsive-load energy than is consumed by the complete fracture of grouted re-bar of equivalent strength, and it remains totally unscathed. Importantly, the results demonstrate that the ejection model is not merely an explanatory concept, but can be used as a representative and valid design method.

Jan 1, 1994

By G. W. Holl, E. G. Fairon

A review of some aspects of shaft design by G. W. HOLL, Pr. Eng., Associate of the Mine Managers Association of South Africa (Fellow)- and E. G. FAIRON, Pr. Eng., F.S.A.I.E.E., M.I. Cert. M.E.E., M.I.Mech. E., (Visitor) The reasons for the choice of vertical shafts in preference to inclined shafts of comparable capacity are set out, and the conclusion that the former are superior in virtually all circumstances is substantiated. The shapes of vertical shafts sunk during the past sixty years are recorded, and the shapes of those sunk during the past decade are analysed. From this analysis it is clear that circular and elliptical (and quasi-elliptical) shafts are the shafts of the future and that the rectangular shape will conceivably become obsolete. The determination of the size of shaft is discussed and it is forecast that shafts of a diameter of 13 metres and over will probably be sunk without difficulty in the years to come. The depth of the shaft is determined intrinsically by the Mines and Works Regulations applicable to factors pf safety of winding ropes and the design of the winding system. The Blair winder is the most suitable winder for hoisting rock, men, and material from a depth of, say, 3000 metres on an economic basis. Tertiary shafts will, therefore, be unnecessary in future deep-level mining projects at the limit of depth now envisaged. A few other factors relating to shaft design are described.

By I. J. Barker

This paper argues that the scale up of furnaces and the supply of electricity are going to be two major issues that will affect the ferro-alloy industry in the near future, and that economic factors will drive this development. The most common type of furnace for producing ferroalloys at present is the submerged-arc furnace with three electrodes, fed from a three-phase AC electrical supply. The scale up of this technology has now reached a fundamental constraint, which is caused by the electrical reactance of the secondary circuit. If the economy of any further scale up is to be achieved in the future then a different technology will have to be used. The supply of electrical power in future is likely to become amore complex issue than it is at present. Existing submerged-arc furnaces tend to run at fairly steady loads, but the ability to swing the load under demand-side management may offer advantages, and may allow a furnace to obtain cheaper power from the organizations that supply this power. This will have to be counterbalanced against the nuisance factors incurred by having a varying load in the operation of the furnace. Various options are therefore discussed in this paper. The possibility of scale up of Søderberg electrodes is discussed, as well as the use of DC power and multiple electrodes. Some of the issues with load swinging are also examined. A particular scenario is also briefly presented to show that considerable further scale up of ferro-alloy furnaces is still possible.

Jan 1, 2011

By S. Potgieter-Vermaak

The main mineral components of the so-called heavy sands (or black sand) are the titanium, zirconium and iron-bearing minerals, rutile, zircon and illmenite, accompanied by various minor and trace amounts of silicates or phosphates, often present together with the rare earth elements. The investigated samples originated from Richards Bay Minerals (RBM) in South Africa, which is jointly owned by BHP Billiton and Rio Tinto, the largest titanium slag producer in the world. The present study focused on the characteristics of samples originating from various points in the separation plant. The heavy-mineral fractions and concentrates of a representative suite of samples were investigated using various analytical techniques to establish the differences in morphological and elemental compositions at the different points in the plant. These techniques included: reflected light microscopy, micro-raman spectrometry (MRS), scanning electron microscopy coupled with energy dispersive X-ray analysis (SEM/EDX) and structural and chemical analyser (SCA). All samples comprised a heterogeneous population of zircon and rutile grains with diverse physico-chemical properties, expressed by large differences in colour and trace element chemistry of single grains. The elemental composition proved the presence of the rare earth elements that appeared to coexist with both Ti and Zr minerals. The percentage of grains hosting inclusions, such as anatase, monazite, quartz, spessartine, augite, epidote and actinolite, varied for each sample and could be uniquely characterized by Raman investigations. Keywords: heavy mineral sands, micro-ram an spectrometry, elemental analysis

By Penswick D.

From the perspective of management control, most mining operations can be represented by a hierarchy comprising four layers. The base of this hierarchy consists of the individual pieces of equipment that are used to break and transport rock. The next level comprises the coordinated interaction between different pieces of equipment, termed a process. An operation results from the bringing together of different processes within a common discipline, such as mining or hydrometallurgy. Finally, an enterprise is formed when the separate mining, concentrating and smelting business units are combined. To date, much of the literature written on the topic of mining automation has focused on the equipment level, specifically on removing the operator from production mining equipment. This focus should not, however, overshadow advances registered in automation at both the process and enterprise levels. Only at the operations level of the hierarchy has little progress been made. While automated equipment, process management and enterprise resource planning are a current reality; it is argued that the benefits from automation have, to date, been largely elusive. It is further argued that full benefits of automating other levels will only be fully achieved when this ‘gap’ in the mining automation hierarchy has been addressed and an uninterrupted, process-wide information flow across the entire operation is achieved. Furthermore, it will be reasoned that so-called vertical integration can only be achieved with an automated operational level in place. It is believed that all the necessary building blocks are in place and that the industry is ready for this development. The paper concludes by presenting a vision for what operational level automation would achieve.

Sep 1, 2001

By T. Gibson

It is generally accepted that South African coal mines have gone through the mechanization of most of its underground mines, and while there certainly were a lot of productivity improvements, the lack of visibility on these machines operating underground still leave vast areas for improvement. The average actual cutting time still remains very low, in the 30 per cent area, which leaves significant improvement possible. Currently all analysis is carried out using hand-written reports. Unfortunately this is subjective and often inaccurate because of human interpretation. In addition in order to avoid costly component failure, a monitoring system is required to warn of eminent failures and condition monitoring. The challenge faced was to supply and install a system that was ‘Open’ and could be retrofitted to existing machines in operation because of the current standard of communication links to the underground sections. In addition it was intended to use, existing ‘off the shelf’ software and hardware and adapting this for the harsh underground mining conditions wherever possible in order to reduce costs. As development occurred, as with all technology, doors opened to other opportunities such as, remote access data logging and Internet based reporting. We focused on the Voest Alpine Bergtechnik (VA) range of continuous miners, as need arose to achieve the desired cost improvements because of the strategy of service contracts the subsidiary company, Voest Alpine Mining & Tunnelling (VAMT) offer the mining industry in South Africa. This paper describes a data logging system that has been installed on various continuous miners that provides data in real time, on line, and that could be expanded to any electrically supplied underground mobile or stationary mining machine.

Sep 1, 2001

By R L. Bowers, D S. Smit

In light of soaring commodity prices, Nkomati has investigated the potential of processing additional low grade nickel ores to add to their reserves. The Chromititic Peridotite Mineralised Zone (PCMZ) was identified as one ore type with the potential of producing a high grade nickel concentrate. In addition to the nickel, copper, cobalt and PGM?s, the ore also contains a significant amount of chromite which adds substantial value to the project. Pyrrhotite, pentlandite, chalcopyrite and chromite are the dominant minerals within the ore, which on average grades around 0.23% Ni and 12% Cr2O3. The PCMZ ore contains excessive amounts of fibrous silicates which makes it a challenging float, with gangue control as the key element to success. Extensive laboratory testwork, which eventually led to a pilot plant campaign, was undertaken with the aim of developing a process route for treating the PCMZ ore. Several flotation circuit configurations were tested and results from the optimal circuit indicated that a nickel concentrate grading at 8.5% is achievable at a recovery of 78%, with substantial recoveries for the copper, cobalt and PGM co-products. The chromite concentrate, grading at 45%, was extracted from the flotation rougher tailings via a spiral plant. These promising results were incorporated into the plant design and are currently being evaluated in a bankable feasibility study, which will see the current 100ktpm MMZ concentrator being upgraded to a 250ktpm PCMZ concentrator as part of Nkomati?s Phase II expansion.

Jan 1, 2007

By M. D. G. Salamon

In view of an urgent need for a simple means of simulating the behaviour of stratified rock masses, a long-dormant linear model is revived, generalized, and discussed in some depth. The model is piecewise homogeneous, consisting of a pile of homogeneous isotropic laminae in which the interfaces between beds are parallel and free of shear stresses and cohesion. The continuity of stresses and normal displacements is assured across the bedding planes and, to maintain simplicity, the theory of thin plates is used as the kernel of the model. The paper covers the case where the rock mass consists of discrete beds in which the layers are arranged either heterogeneously or systematically. The stratification is regarded as uniform when all the layers are of the same thickness and their Young's modulus changes exponentially with depth; and as homogeneous when all the laminae have identical thickness and properties. The case of uniform stratification is also treated on the basis of an equivalent quasi-continuum derived from the basic model. Attention is focused on the derivation of influence functions. It is demonstrated that the quasi-continuous models, when they are available, can replace their distinct-layer counterparts. The homogeneous quasi-continuum, which permits the derivation of its influence functions in closed form, facilitates the determination of the roof and floor convergence. In the case of a significant class of problems, the convergence distribution is defined by a second-order differential equation in the xy plane, which simplifies to a Poisson's equation if the mining excavations are unsupported. With this result, the paper prepares for the solution of a host of practical problems. These solutions are presented in a series of papers that are in the course of publication.

Jan 1, 1991

By Q. G. Reynolds

The problem of direct-current plasma arc behaviour, interaction, and dynamics is considered in the context of metallurgical DC arc furnace applications. Particular attention is paid to the transient flow behaviour of arc systems. A mathematical formulation of the physics used to describe the arc system is presented, and includes the spatial and temporal evolution of fluid flow, heat transfer, and electromagnetism. Based on this formulation, a numerical model is developed using a finite difference approach on a regular cartesian grid in both two and three dimensions, with a special focus on robust stability, high resolution modelling, and high performance. A collection of results produced using the numerical model to study pilot plant-scale furnaces is then presented. These address a range of process and design variables and their effect on the numerical model?s results. Where possible, the qualitative behaviour of the model is compared to available experimental data. A number of novel effects and phenomena are seen in the dynamic behaviour of the DC plasma arc model for both single and multiple arc systems, which may lead to improved understanding, control, and manipulation of such systems where they occur in industrial applications. Keywords Pyrometallurgy, direct-current furnace, plasma arc, modelling.

Jan 1, 2010

By J. H. Bulloch

A study of fatigue-crack growth in a rail steel and an associated weld metal showed that fatigue cracks in the deformed rail steel had slightly higher growth rates than in undeformed rail steel. The crack growth in the weld metal was appreciably faster than either and exhibited plateaux that were above the upper bound reported for rail steels, The cyclic-life curves showed that, over an indicated service period of 8 years, both the deformed and the undeformed rail steels should not be prone to fatigue failures under the defined mechanical conditions, For the weld metal, the cyclic-life curves indicated the possibility of premature fatigue failures but, as al the input parameters represented a 'worst case' situation, this assessment could wel be ultra-conservative. In the rail steels at low levels, the extension of the fatigue cracks occurred by ductile striated growth. At a Kmax level of 35 MPa .J/n, cleavage facets formed initially on the surface of the fatigue fractures, and their incidence increased with Kmax until the onset of fast failure, The microstructure of the deformed rail steel exhibited cleavage facets at a slightly lower Kmaxvalue, and contained more cleavage facets at a given K level, than the microstructure of the undeformed rail steel. The average size of the cleavage facets agreed well with the pearlite nodule dimensions of 60 to 100 ILm. The microstructure of the weld metal was much coarser and contained columnar grains. At al f.K levels, the dominant fracture mode was cleavage, with small isolated regions of ductile striated fatigue cracks. The size of the cleavage facets varied from 150 to 600 ILm, and such a variation may be explained by the fact that cracking generally occurs in association with the pro-eutectoid ferite phase, and that this phase is distributed along some boundaries of prior austenite grains and the boundaries of the coarser columnar grains.

Jan 1, 1987

By T. Tholana, P. N. Neingo

"A mineral resource constitutes the principal underlying asset of a mining company, and must be exploited and managed in such a way that maximum value is derived from it. Various asset management frameworks applicable to physical assets are available. This paper focuses on the extension of an asset management approach, particularly the Publicly Available Specifications (PAS) 55 asset management framework and the International Organization for Standardization (ISO) 55000 series of standards, to mineral assets. It is concluded that PAS 55 and the ISO 55000 series of standards can be extended to manage mineral assets, resulting in an integrated approach to sustainably optimize value from the company’s mineral assets. The benefits include improved returns on mineral assets, maximum mineral asset utilization, creation of an organizational culture focused on quality and continuous improvement, and assurance to stakeholders that the mineral assets are being efficiently managed over their entire life-cycle. In the context of this paper, a ‘mineral resource’ refers to both Resources and/or Reserves as classified by the SAMREC Code. IntroductionMining involves the extraction of valuable mineral resources from the Earth’s crust. A mineral resource is unique among natural resources in the sense that it is a wasting asset, which means that its value depreciates continuously with extraction. In addition, it is often impaired from time to time as commodity price changes (Macfarlane, 2011a). Since a mineral resource is a wasting asset there is a single opportunity available to optimize value from its extraction. Mine planning involves identifying a strategy to exploit the mineral resource in a way that maximizes value at an acceptable risk level throughout the life of mine (LOM). This optimum extraction strategy changes as economic conditions change, particularly commodity prices, as well as with improvements in orebody knowledge as mining progresses. It is the mineral deposit that competitively distinguishes two different mines and dictates the location of a mine; hence the adage ‘the orebody dictates’. Therefore, a mineral resource is the primary tangible asset possessed by mining companies and must be managed in such a manner that maximizes its value at an acceptable risk level (Macfarlane, 2011a)."

Jan 1, 2016

By J. Githiria, C. Musingwini, J. Muriuki

"The maximization of net present value (NPV) is a primary objective in open pit mine planning processes. In an attempt to meet this objective, cut-off grades are considered in all the stages of mining. There are three stages involved in resource extraction, namely mining, processing, and refining/marketing, which are all defined in Lane’s approach. Using Lane’s approach, the economics involved in each stage are identified independently and interact to provide an optimum cut-off grade.A hypothetical block model is used to illustrate how a computer application (Cut-off Grade Optimiser) that was developed in this study is used in the optimization of cut-off grades using Lane’s algorithm. This paper analyses the application of Lane’s approach for a single element in cut-off grade optimization as applied in the calculation of the optimum cut-off grade through a computer-aided application.Although Lane’s approach is not complex it is not widely applied to maximize the NPV of mining operations as the iterative calculations can be lengthy. The computer application developed in this study shows how Lane’s algorithm combined with a linear programming function can optimize cut-off grade with regard to the complex situations faced by mining operations. The application saves users from performing otherwise lengthy, iterative calculations.Background and previous researchAn open pit mining complex constitutes multiple pits, ore processing streams, and metal refining options. An open pit mine produces ore (valuable) and waste material. Ore is transported to an appropriate processing stream for producing concentrate. A refinery processes the concentrate to produce marketable metal product (Lane, 1964, 1988).Cut-off grade optimization is based on Lane’s theory, where a mining system consists of three major stages such as mining, milling, and refining. In this approach the cut-off grade selection is done between six cut-off grades that involve all stages of a mining operation. Selection of the optimum cut-off grade influences the net present value (NPV) by maximizing the net cash flows from the operation."

Jan 1, 2016

By J. Sjöberg, P. Lundman, E. Nordlund, C. Quinteiro

Abstract In the Kiirunavaara underground mine, crude ore is transported through ore passes from the production levels to the main haulage level. Failure and stability problems have occurred in some of these ore passes, with increasing frequency as mining depths increase. In this paper, selected results are presented from a research project run by LKAB. The presentation focuses specifically on stability analysis. Numerical models were used to identify failure mechanisms, in conjunction with empirical data and engineering judgment. The most important controlling factors governing failure and damage in ore passes were quantified. A causative model was established in which failure mechanisms and failure development were described. This model was used to formulate a prognosis of future ore pass stability, and for recommending remedial measures. Résumé Le transport du minerai entre les niveaux de production et le niveau principal de transport, à la mine de fer de Kiruna, se fait par des passes à minerai. Avec la profondeur d'extraction grandissante, les problèmes de stabilité au niveau de ces passes à minerai se sont multipliés. Les résultats d'un programme de recherche sur la stabilité des passes à minerai à la mine de fer de Kiruna sont présentés dans cet article. Les mécanismes de rupture ont été identifiés à l'aide de modèles numériques, de données de terrain et du jugement empirique des ingénieurs. Les facteurs principaux controllant l'instabilité des passes à minerai ont aussi été quantifiés. Un modèle a été établi dans lequel les mécanismes de rupture et le développement des dommages ont été décrits. Ce modèle a été utilisé à fins prédictives et remédiatives. Zusammenfassung In der Grube Kiirunavaara wird das gewonnene Erz über Sturzschächte von den Abbauebenen zu der Hauptförderebene gefördert. Mit zunehmender Tiefe des Abbaus sind vermehrt Standsicherheitsprobleme in den Sturzschächten aufgetreten, die teilweise zu Verbrüchen geführt haben. In dem vorliegenden Beitrag werden ausgewählte Ergebnisse einer Forschungsarbeit des LKAB präsentiert. Dabei wird der Schwerpunkt auf die Standsicherheitsuntersuchungen gelegt. Zur Ermittlung der Bruchmechanismen wurden numerische Modelle in Verbindung mit empirischen Daten angewendet. Die wichtigsten Einflußfaktoren für das Versagen der Sturzschächte wurden ermittelt. Es wurde ein Modell zur Beschreibung der Bruchmechanismen und des Versagensablaufs aufgestellt. Dieses Modell wurde verwendet, um die Standsicherheit zukünftiger Sturzschächte zu prognostizieren und um Sanierungsmaßnahmen zu erarbeiten.

Jan 1, 2003

[The following notations will be used: O an ore body. W a sample of size w located in O; the size of this sample. W a block or a sample, of size W, located in O; the size of this block or sample. z' a point in W. W' a block or a sample, of size W' greater than or equal to W, located in O; the size of this block or sample. z" a point in W'. y (h) point semivariogram in O. s (h) point covariogram in O. In the previous chapter, we saw that the point semivariogram y (h) usually cannot be estimated directly from the data available. Rather, the values of samples of size w are available in O, and from these a regularized semivariogram y w (h) can be estimated. The following relationship has been proved (5.8): Y w(h) = Y (w;w+h) - Y (w;w). y (w;w+h) is the average value of y (z' z") when z' takes all possible positions in a sample w and z" all possible positions in a sample w+h of the same size, h being the distance between w and w+h. y (w;w) is the value of y (w;w+h) for h = 0, i.e. the average value of y (z' z") when z' and z" take all possible positions in w. We also saw that the regularized covariogram aw(h) is related to the point covariogram a (h) as follows (5.9): w(h) Gr In the following chapters, it will be shown that calculation of the average value y of y, or -a of a, in the general case relating to samples or blocks W and W' which need not be of the same size, plays a very important part in all the practical applications of geostatistical theory. The purpose of this chapter is to introduce various methods of calculating y and a-. Furthermore, mathematical equations and graphs are provided which can be used when the point semivariogram can be represented by one of the more common equations, and when the blocks W and W' are of simple shape. By definition (Fig. 6.1): [ ] Y (W;W) = WW f dz' f y (z' z") dz" (6.1) in W z" in W']

Jan 1, 1978